Chris, this is just my opinion, but you need at least a 400mm lens if you want to go birding, and that costs. Expect to spend close to four figures if you want decent performance at these focal lengths.
I'm a Canon shooter and I'm thinking in terms of Canon lenses and then translating into Nikon equivalents, but my first recommendation would probably be the Nikon 80-400 VR, particularly if your interest is more in perched/standing birds, rather than birds in flight. The only big drawback with the 80-400 is that it's not AF-S, and some folks perceive the AF speed to be slow. But then, when you're shooting BiF, any autofocus speed seems slow.
If, however, you're happy with the reach of the Tamron 70-300 and you just want better quality, then upgrading to the Nikkor 70-300 VR might be an option.
If you can't afford the 80-400VR, there are the Sigma alternatives, like the 120-400 OS HSM and 150-500 OS HSM, and they're great value. But. I once read
a lensrental piece where they said both of those lenses were in their "Hall of Shame" for a 45% failure rate. They may have gotten a bad batch, and rentals are obviously going to see a lot more (ab)use than the average, but it's still something to consider.
And, on top of that, as a Canon shooter, I've seen
a review/test on the juzanature.com site, which compared the 120-400 OS, 150-500 OS, the old Sigma 50-500 (aka "the Bigma") and the Canon 100-400L, and of the three Sigmas, the Bigma was actually the best performer. The only trouble is that the Bigma is also a beast to handhold. Once you go past 400mm, you'll probably want to be shooting with a tripod or at least a monopod.
In terms of prime vs. zoom for supertele--it depends on the usage. A prime is typically going to take teleconverters better and autofocus faster than a zoom. But I seriously curse my 400/5.6L USM prime when I'm at the zoo (and I run backwards a lot). A 400mm prime can be very limiting, unless
everything you want to shoot is too far away, anyways. Which pretty much sums up birding photography.
One other small note. With the Tamron, if you haven't, try stopping it down to f/8 or f/11 and see if that helps you with sharpness. A lot of the lower-cost 70-300s have lousy wide-open performance, but can improve when stopped down.